174
ANALYTICAL MECHANICS
Thus Young's modulus of a substance equals, numerically, the force necessary to stretch a uniform rod of unit cross-section, which is made of the given substance, to double its length. During the process of stretching Hooke's law is, of course, supposed to hold.
139. Work Done in Stretching an Elastic String. — Let L denote the natural length of the string and x its length at any instant of the process of stretching. Then the work done in increasing the length by dx is
dW=Tdx = ASdx,
where T is the tensile force, S the tension, and A the area of the cross-section of the string. But by Hooke's law,
In this case s =
and
FIG. 95.
where X7 = A\y and I is the total increase in length. Thus the work done varies as the square of the increase in length. Plotting I as abscissa and W as ordinate we obtain a parabola, Fig. 95.
140. Work Done in Compressing Fluids. —Let C, Fig. 96, be a cylinder which contains a compressible fluid and which is provided with a piston. When the piston is displaced toward the left work is done against the force with which the fluid presses upon the piston. If dx denotes the dis-